What is the base of our numerical system?

Our numerical system is a decimal or base ten system. It uses digits from 0 to 9 as a base.

Our numerical system is a place-value system. This means that the place or location of a numeral determines its numerical value.

subsets of real numbers

The following are subsets of real numbers:

- Natural numbers
- Whole numbers
- Integers
- Rational numbers
- Irrational numbers

**natural **numbers

**Natural numbers** are the set of counting numbers.

{1, 2, 3, 4, 5...}

Natural numbers are comprised of odd and even numbers.

The smallest natural number is 1; the largest natural number is infinity.

**whole **numbers

**Whole numbers** are the set of natural (counting) numbers *and zero*.

{0, 1, 2, 3, 4, 5...}

Whole numbers are comprised of odd and even numbers.

**integers**

**Integers** are the set of *natural* numbers, their *negative* opposites, and *zero*.

{...-3, -2, -1, 0, 1, 2, 3...}

Integers are comprised of whole numbers and the opposites of natural numbers.

**rational **numbers

**Rational numbers** are the numbers that can be expressed as simple fractions of two integers -- i.e. as ratios.

*** The denominator in the fraction cannot be zero.

*Examples: *

5 = ^{5}*/*_{1} 1.75 = ^{7}*/*_{4}

Rational numbers consist of integers and non-integral numbers (numbers that have terminating or repeating decimals).

**irrational **numbers

**Irrational numbers** are the numbers that *cannot* be written as terminating or repeating decimals.

*Example: *

For the purposes of the SAT, the most important irrational numbers are the square root of 2, the square root of 3, and Pi.

**even **numbers

A number that is divisible by 2 is called an **even number**.

{...-4, -2, 0, 2, 4...}

All numbers ending in 0, 2, 4, 6, and 8 are even.

**odd **numbers

A number that is not divisible by 2 is called an **odd number**.

{...-5, -3, -1, 1, 3, 5...}

All numbers ending in 1, 3, 5, 7, and 9 are odd.

Is the sum of two even numbers even or odd?

EVEN + EVEN = ?

EVEN + EVEN = EVEN

*Example*:

10 + 2 = 12

Is the difference between two even numbers even or odd?

EVEN - EVEN = ?

EVEN - EVEN = EVEN

*Example:*

10 - 2 = 8

Is the sum of two odd numbers odd or even?

ODD + ODD = ?

ODD + ODD = EVEN

*Example*:

5 + 5 = 10

Is the difference between two odd numbers odd or even?

ODD - ODD = ?

ODD - ODD = EVEN

*Example:*

5 - 3 = 2

Is the sum of an odd number and an even number odd or even?

EVEN + ODD = ?

EVEN + ODD = ODD

ODD + EVEN = ODD

*Examples:*

4 + 3 = 7

5 + 4 = 9

Is the difference between an odd number and an even number odd or even?

EVEN - ODD = ?

ODD - EVEN = ?

EVEN - ODD = ODD

ODD - EVEN = ODD

*Examples*:

6 - 5 = 1

7 - 2 = 5

Is the product of two even numbers odd or even?

EVEN x EVEN = ?

EVEN x EVEN = EVEN

*Example:*

6 x 8 = 48

Is the product of two odd numbers odd or even?

ODD x ODD = ?

ODD x ODD = ODD

*Example:*

3 x 7 = 21

Is the product of an odd number and an even number even or odd?

EVEN x ODD = ?

EVEN x ODD = EVEN

*Example*:

6 x 3 = 18

*** When dividing odd or even numbers, the result can be a fraction, which is not a whole number; therefore, it is neither even nor odd.

When you raise even numbers to odd powers, is the result odd or even?

(EVEN)^{ODD} = ?

(EVEN)^{ODD} = EVEN

*Example*:

2^{5} = 32

When you raise even numbers to even powers, is the result odd or even?

(EVEN)^{EVEN} = ?

(EVEN)^{EVEN }= EVEN

*Example*:

4^{4} = 256

When you raise odd numbers to odd powers, is the result odd or even?

(ODD)^{ODD} = ?

(ODD)^{ODD} = ODD

*Example*:

3^{3} = 27

When you raise odd numbers to even powers, is the result odd or even?

(ODD)^{EVEN} = ?

(ODD)^{EVEN }= ODD

*Example*:

7^{2} = 49

True or False?

Any operation (addition, subtraction, multiplication, division or raising to power) on *even *numbers with another even number will result in an *even *number answer.

True.

If you understand that any two even numbers are divisible by 2, then logically the sum, the difference, the product, the quotient, the power of the two will always be divisible by two.

True or False?

Any operation (addition, subtraction, multiplication, division or raising to power) on *odd *numbers will result in an *odd *number.

False.

- The sum and the difference of two odd #'s are even
- The product, the quotient, and the power is odd

Think of ODD numbers as EVEN + 1. Or remind yourself that odd numbers end in 1, 3, 5, 7, or 9.

*Example: *

ODD + ODD = EVEN + EVEN + 2 = EVEN.

How do you express an odd number in terms of an even number?

ODD = EVEN + 1

*Example: *

ODD + EVEN = EVEN + EVEN + 1

How should you use number facts like ODD + ODD = EVEN on the SAT test?

You don't have to memorize them, but you have to be able to see that some questions may need you to recall and connect these facts to solve quickly.

Remember, SAT type questions often use simple facts, and the trick is seeing through them for a quick solution.

What are **consecutive **numbers?

**Consecutive **numbers follow the natural order and differ by 1.

{...4, 5, 6, 7, 8, 9...}

Consecutive even and consecutive odd numbers differ by 2.

{...2, 4, 6, 8...}

{...3, 5, 7, 9...}

In a set of *consecutive *integers, how do you find the number of integers between the smallest and the largest numbers, inclusively?

To count consecutive integers, *subtract *the smallest from the largest and *add 1*.

*Example*:

Count the integers from 14 to 51.

54 - 14 + 1 = 41

What type of number do you get as a result of adding *different consecutive positive odd *numbers?

1 + 3 = ?

1 + 3 + 5 + 7 + 9 = ?

The sum is a *perfect square *of the number of numbers being added together.

1 + 3 = 4 = 2^{2}

1 + 3 + 5 + 7 + 9 = 25 = 5^{2}

**prime **numbers

A **prime number** is a natural number *greater than 1* whose only factors are *itself *and *1*.

The following is a set of prime numbers less than 100:

{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97}